Types of modeling methods and the corresponding objects are known. The reader is referred for example to the applicant's DE-B4-101 29 565 and especially to DE-B3-102 51 716.
In process automation of the handling of steel the modeling of the metal temperature has particular significance. In such cases it is basically of no consequence whether the modeling of the metal temperature is undertaken in relation to a steel volume which is located in an oven or is passing through a rolling process or a cooling process for example. Phase transitions which can also occur with metals within the fixed phase make computation with the thermal conductivity equation difficult in such cases. If an accurate computation is required, the phase transition must be included in the determination of the time gradient of the state of the metal volume.
With steel in particular the phase transition only occurs after a delay since the temperature changes occurring are so large that the transition cannot follow the temperature change. Frequently this is caused by alloying elements which are added to the steel.
The reason for this delayed transition lies—in steel for example—in the presence of carbon in the steel. This is because, although carbon dissolves relatively well in austenite, it only dissolves to a very small, practically negligible extent in ferrite. This delays phase transitions from austenite to ferrite since the carbon in the austenite must first diffuse out before ferrite can form.
Austenite can also form a further phase, namely cementite. Cementite together with ferrite forms a mixed phase which is also referred to as pearlite. At first sight it appears completely unclear and almost hopeless to take such a complicated process into account correctly in a thermal conductivity equation.
In the prior art the following solutions have been initially attempted:                The modeling of the phase transition has been greatly simplified.        The modeling of the phase transition was undertaken in an advance computation with an approximated temperature gradient. The phase transition was then fixed. Exothermic processes in the phase transition were taken into account by heat sources in the Fourier thermal conductivity equation. However this coupling of the phase transition in the form of heat sources to the Fourier thermal conductivity equation only appears to solve the problem. A more precise consideration actually shows that the approach is physically incorrect. This is especially evident from the fact that parameters are to be set separately for the heat sources, which is not required for a correct solution.        
An already significantly improved approach is known from DE-B4-101 29 565, in which the thermal conductivity equation is correctly applied. The phase transition equation is however only valid for a two-phase system (e.g. the austenite-ferrite system). An expansion to a three-phase system (e.g. the austenite-ferrite-cementite system) is not easily possible. Also the variables thermal conductivity and temperature occurring there in the thermal conductivity equation are only dependent on the enthalpy and a phase component. The dependence on a phase component is sufficient in such cases since because of the observation of a pure two-phase system there, the second phase component is implicitly produced from the situation that the sum of the two phase components must be one.
In DE-B3-102 51 716, also cited above, the thermal conductivity and the temperature are functions which depend on the enthalpy and the components of all phases considered. In a three-phase system, e.g. the austenite-ferrite-cementite system mentioned in DE-B3-102 51 716, the thermal conductivity and the temperature are thus functions which depend on the enthalpy and all phases.
Attempts to implement the two last-mentioned steps have actually led to significantly better results than previously. However discrepancies have occurred between the behavior of the steel volume determined in accordance with the modeling method and the actual behavior of a corresponding actual steel volume.